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Journal of Physics A Mathematical and General

Quantum mechanics as classical mechanics plus quantum corrections: the cubic anharmonic oscillator

Gabriel Álvarez

Published Mar 21, 1989• 12 citations
Publisher

Abstract

The Rayleigh-Schrodinger perturbation series for the cubic anharmonic oscillator can be formally rearranged as the classical Birkhoff series plus quantum corrections proportional to successive powers of h(cross) 2 . Each of the individual quantum corrections is a convergent power series expansion (with the same radius of convergence as the classical series) of the corresponding term in the Jeffreys-Wentzel-Kramers-Brillouin series.

DOI
10.1088/0305-4470/22/6/014

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The Rayleigh-Schrodinger perturbation series for the cubic anharmonic oscillator can be formally rearranged as the classical Birkhoff series plus quantum corrections proportional to successive powers of h(cross) 2 . Each of the individual quantum corrections is a convergent power series expansion (with the same radius of convergence as the classical series) of the corresponding term in the Jeffreys-Wentzel-Kramers-Brillouin series.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
The detailed transition from classical mechanics to quantum mechanics, particularly in regard to the question of quantum corrections to the semiclassical quantization, can be completely understood for the x4{x}^{4}x4 perturbed harmonic oscillator. This is a consequence of five facts: (i) The quantum perturbation series for the energy can be solved order by order in closed form. (ii) The quantum series rearranges directly into the classical canonical perturbation series plus quantum corrections proportional to successively higher powers of ${\ensuremath{\Elzxh}}^{2}$. (iii) The classical series and the subseries for the quantum corrections converge (subseries by subseries) to the terms of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) expansion for the energy (to all orders of \ensuremath{\Elzxh}). (iv) The (all-order) JWKB expansion, conversely, yields the quantum perturbation series and consequently uniquely determines the exact eigenvalue. (v) The ratio of the quantum energy to the classical energy at action (n+1/2)\ensuremath{\Elzxh} differs from 1 asymptotically by 1/[9\ensuremath{\pi}(n+1/2)2{+1/2)}^{2}+1/2)2], independent of both \ensuremath{\Elzxh} and the perturbation coupling constant, as the harmonic-oscillator quantum number n approaches \ensuremath{\infty}. An interesting additional feature is the mechanism by which the convergence of the classical perturbation series and of the series for the quantum corrections gets turned into the divergence of the quantum perturbation series.
Exactly solvable quantum systems are rare—the harmonic oscillator, the hydrogen atom, the square well—and the physics of everything else is the physics of approximation. This article presents a narrative review of the primary literature that built quantum mechanics' approximation methods, from Rayleigh's Theory of Sound, whose treatment of slightly perturbed vibrating systems gave perturbation theory its name and scheme, and Ritz's 1909 variational method, which converted boundary-value problems into energy minimizations, through Schrödinger's 1926 wave mechanics, which equipped the new quantum theory with perturbation theory from the start, Born and Oppenheimer's 1927 adiabatic separation of electronic and nuclear motion, the 1926 papers of Wentzel, Kramers, and Brillouin that independently derived the semiclassical expansion bearing their initials, Hartree's 1928 self-consistent field and Hylleraas's 1929 variational computation of helium, which made the many-electron atom calculable, Kato's 1949 mathematical foundation of the perturbation method's convergence, Landau and Lifshitz's codifying textbook, and Bender and Wu's 1969 demonstration that the anharmonic oscillator's perturbation series diverges—the asymptotic character that defines the method's limits. The synthesis is organized around three themes: the perturbation program, in which small parameters turned the solvable cases into engines for the unsolvable; the variational program, in which upper bounds and trial functions turned minimization into the many-body atom's working instrument; and the semiclassical bridge, in which the WKB expansion connected wave mechanics to action-angle quantization and exposed tunneling. It is concluded that the three programs are complementary solutions to one problem—the gap between what can be solved and what must be calculated—and that their asymptotic character, made precise by Bender and Wu, is not a defect but the field's realistic self-understanding.
Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.
A relation between classical canonical transformations and quantal unitary transformations is investigated with the use of the phase space method. A unitary transformed operator expressed in the phase space is expanded into a power series of ħ, which shows that a canonical transformation can be regarded as a classical approximation (the limit ħ→0) to the unitary transformation. The expansion provides us with a procedure to successively evaluate quantum corrections starting with the classical limit. As an illustrative example, the lowest-order quantum correction to the Birkhoff-Gustavson transformation for a one-dimensional anharmonic oscillator is calculated in the case of Weyl correspondence rule.